The spectral division method (SDM) is one of several sound field synthesis techniques of spatial sound reproduction. The synthesis accuracy of SDM is usually higher than that of wave field synthesis (WFS). However, existing SDM-based sound field synthesis methods are developed for virtual sources moving uniformly along a straight trajectory. This Letter proposes a 2.5-dimensional SDM driving function for arbitrary moving sources. Numerical simulations have validated the driving function. The results indicate that the synthesis error of the SDM with the proposed driving function is lower than that of WFS, especially when the virtual source is close to the secondary sources.

## 1. Introduction

Sound field synthesis^{1,2} aims to generate a desired target sound field physically over a synthesis area utilizing multiple loudspeakers. Recently, sound field synthesis of moving virtual sound sources^{3–8} has attracted extensive attention of researchers and engineers. Wave field synthesis (WFS)^{9,10} is a technique for sound field synthesis. Based on 2.5-dimensional (2.5D) WFS, the sound field synthesis of moving virtual sources with arbitrary trajectory and velocity has been realized.^{8} However, 2.5D WFS could be invalid when the virtual source moves in an area near the secondary source distribution (SSD). The reason is that the 2.5D WFS driving function is derived under the high-frequency/far-field assumptions of the virtual source and the SSD. The spectral division method (SDM)^{11,12} is another technique for sound field synthesis. Compared with WFS, SDM is more accurate and remains valid even when the virtual source is extremely close to the SSD.^{13}

To the authors' knowledge, with existing SDM-based sound field synthesis techniques, only virtual sources moving uniformly along a straight trajectory^{5,6} have been investigated so far. However, in practice, most sources move on complicated trajectories, which demands the development of an SDM-based sound field synthesis method for arbitrary moving sources. The present work aims to derive a general 2.5D SDM driving function for the sound field synthesis of moving point sources. This driving function can be applied to moving virtual sources with arbitrary trajectory and subsonic speed in the synthesis plane, and does not introduce a restriction of the distance between the virtual source and the SSD. The time-domain driving function is derived in the present work, and then validated by numerical simulations.

## 2. Sound field of arbitrary moving sources

An arbitrary sound source can be decomposed into a superposition of point sources, and thus the sound field synthesis of the moving point source is considered. As shown in Fig. 1(a), let the position and velocity of a moving point source be $xs(t)=[xs(t),ys(t),zs(t)]T$ and $vs(t)=(dxs(t)/dt)$, respectively. Let the sound wave be radiated by the moving point source at time $t\u2212\tau (x,t)$ and reach the receiving point **x** at time *t*, where $\tau (x,t)$ is the propagation time-delay defined by $|x\u2212xs(t\u2212\tau (x,t))|=c\tau (x,t)$ and $c=343\u2009m/s$ is the speed of sound.

The sound pressure at the receiving point **x** can be written as

where *q*(*t*) denotes the radiation intensity of the source and $g(x,t)=\delta (t\u2212|x|/c)/(4\pi |x|)$ denotes the retarded Green's function. Using the property of the Dirac function $\delta (x)$ yields the target sound field of the moving point source^{8}

where $\Delta (x,t\u2212\tau )=|x\u2212xs(t\u2212\tau )|\u2212(1/c)\u27e8vs(t\u2212\tau )\xb7(x\u2212xs(t\u2212\tau ))\u27e9$ and $\u27e8\xb7\u27e9$ denotes inner product.

## 3. 2.5D SDM driving function of moving sources

### 3.1 2.5D SDM

The use of SDM aims to generate a sound field that coincides with the target sound field radiated by a moving virtual point source. In most application scenarios, the moving virtual point source and the listeners are in the same plane, termed as the synthesis plane. Considering that it is usually desired that the sound field in the synthesis plane is synthesized correctly, sound field synthesis using the linear SSD is sufficient as long as the amplitude decay deviation of the synthesized sound field is acceptable. The approximation of using the linear SSD to synthesize sound field in the synthesis plane is termed as 2.5D sound field synthesis. As shown in Fig. 1(b), an infinite continuous straight-line SSD located along the *x* axis is used to synthesize the sound field radiated by the virtual point source moving in the half-plane where $ys(t)<0$ and $zs(t)=0$. The synthesized sound field can then be written as

where $x0=[x0,0,0]T$ is the position of the SSD and $d(x0,t)$ is the time-domain driving function of the SSD. Note that the integral in Eq. (3) is a convolution of *d*(*x*, *t*) and *g*(*x*, *t*). Applying the spatio-temporal Fourier transform to Eq. (3) yields $P\u0303(kx,y,z,\omega )=D\u0303(kx,\omega )G\u0303(kx,y,z,\omega )$, where $P\u0303(kx,y,z,\omega ),\u2009D\u0303(kx,\omega )$, and $G\u0303(kx,y,z,\omega )$ are the spatio-temporal spectra of $p(x,t)$, *d*(*x*, *t*), and *g*(*x*, *t*), respectively. Similar to previous works,^{5} the synthesized sound field along a reference line $y=yref$ (where $yref>0$) is considered, and it follows that

and the time-domain driving function can be obtained by applying the inverse spatio-temporal Fourier transform to $D\u0303(kx,\omega )$. This method for calculating the driving function is called 2.5D SDM.

### 3.2 The driving function

Using Euler's formula and some integrals (Ref. 14, 3.876), the spatio-temporal Fourier transform of $g(x,t)$ is

where $j=\u22121,\u2009rt=y2+z2,\u2009k=\omega /c,\u2009H0(1)(x),\u2009H0(2)(x)$, and $K0(x)$ are the zeroth-order Hankel function of the first kind, the zeroth-order Hankel function of the second kind, and the zeroth-order modifed Bessel function, respectively. The spatio-temporal Fourier transform of Eq. (1) derives to

Substitution of Eqs. (5) and (6) into Eq. (4) yields the spatio-temporal spectrum of the driving function

Under high-frequency/far-field assumptions, i.e., $|kx2\u2212k2||yref\u2212ys|\u226b1$ and $|kx2\u2212k2||yref|\u226b1$, the large argument approximations of the Hankel function and the modifed Bessel function can be applied to Eq. (7), which yields

where $S(kx,\omega )$ is equal to *j* (for $k2\u2212kx2>0$ and *k* > 0), –*j* (for $k2\u2212kx2>0$ and *k* < 0), or 1 (for $k2\u2212kx2<0$). Using some integrals (Ref. 14, 6.677), the inverse spatial Fourier transform of Eq. (8) yields the frequency-domain driving function

where $xs=xs(t\u0302)$ and $ys=ys(t\u0302)$. It is worth noting that, for the case that the moving virtual point source is a static point source, i.e., $xs$ and $ys$ do not change with time, Eq. (9) is reduced to the driving function obtained in Ref. 13.

Applying the inverse temporal Fourier transform to Eq. (9) finally yields the time-domain 2.5D SDM driving function in the form which contains a single integral (refer to the supplementary material for the detailed derivation^{15}), i.e.,

where

and $\Delta t$ [$0<\Delta t\u226a1$ and $\Delta t\u226a\tau (x0,t)$] is a constant introduced in the derivation of Eq. (10).

The integral in Eq. (10) cannot be further simplified due to the arbitrariness of the motion of the virtual point source. Equation (10) is a general equation for the 2.5D SDM driving function, and a more practical equation will be derived below. The practical driving function can be calculated by numerical method. Note that Eq. (10) is still valid when the moving virtual point source moves in the near field of the SSD. The reason is that the derivation of Eq. (10) only requires that the receiving point is far away from the moving virtual point source and the SSD, and does not introduce the restriction of the distance between the moving virtual point source and the SSD.

For numerical calculation, it is a difficult task to determine $\Delta t$ if Eq. (10) is used to calculate the time-domain driving function. The condition $0<\Delta t\u226a1$ should be satisfied but a smaller $\Delta t$ will lead to a larger truncation error. In order to address this problem, consider the moving source that starts radiating the sound wave at time *t*_{0}, and then $1/\Delta t$ can be equivalently expressed as

Equation (12) is more practical for numerical calculation because the problem of choosing an appropriate $\Delta t$ is avoided and the integral interval is finite.

## 4. Numerical examples and comparison

Two examples are presented in order to validate the proposed SDM driving function. In example 1, a virtual point source moves on a circular trajectory given by $xs(t)=[sin\u2009(200t),\u2009cos\u2009(200t)\u22121.05,0]T$, and in example 2, a virtual point source moves on a cosine trajectory given by $xs(t)=[100t,0.5\u2009cos\u2009(400\pi t)\u22120.55,0]T$. In both examples, the moving virtual point sources oscillate in pressure at a frequency of $500\u2009\u2009Hz$. To simulate a continuous SSD, an array of secondary point sources is linearly distributed along the *x* axis with a source-to-source spacing of $0.02\u2009m$ and a total length of $30\u2009m$. A reference line, *y* = 1, is investigated in both examples.

The sound field radiated by the moving virtual point source (referred to as the target sound field) is calculated by using Eq. (2). The sound field synthesized by the SSD (referred to as the synthesized sound field) is then calculated for comparison with the driving function determined by using Eq. (12). Figure 2 shows the simulation results for *t* = 0. As shown in Figs. 2(c) and 2(f), the relative difference between the target and synthesized sound fields (referred to as synthesis error) is small in each example, indicating the validity of the proposed driving function. It is worth noting that the synthesis error reaches a minimum along the reference line, which is consistent with the stationary case.

In order to better compare the proposed SDM driving function and the WFS driving function, the time histories of the sound pressure and the synthesis error are calculated for the two examples. The WFS driving function is based on Ref. 8. Figure 3 shows the time histories of the sound pressure and the synthesis errors calculated by SDM and WFS, respectively. Frequency variation associated with the motion of the virtual point source can be seen clearly in Figs. 3(a) and 3(d), which is termed as the Doppler effect. As shown in Figs. 3(b), 3(c), 3(e), and 3(f), the synthesis error of SDM is less than that of WFS. For example 1, the distance between the moving virtual point source and the SSD reaches a minimum approximately at $t=0.330\u2009ms$. Correspondingly, the sound wave radiated by the moving virtual point source at $t=0.330\u2009ms$ arrives at the receiving point $[0,1,0]T$ at $t=3.403\u2009ms$ when the synthesis error of WFS reaches a maximum. However, the synthesis error of SDM is still quite low, as shown in Fig. 3(b). Similar phenomenon can be observed in Fig. 3(e) (for example 2). This phenomenon might indicate that, for WFS, the synthesis error increases when the moving virtual point source is close to the SSD, whereas it is not the case for SDM. It may be caused by the elimination of the high-frequency/far-field condition between the virtual source and the SSD for deriving the SDM driving function.

## 5. Conclusion

In summary, this Letter proposes a 2.5D SDM driving function for the sound field synthesis of arbitrary moving virtual sources. Compared with WFS, the high-frequency/far-field condition between the virtual source and the SSD is eliminated with the proposed driving function. The method yields better sound field synthesis performance, especially when the virtual source is close to the SSD. The time-domain SDM driving function is then derived in the form that contains one single integral. The proposed driving function generalizes SDM to arbitrary moving virtual sources. The validity of the proposed driving function has been verified by numerical simulations of two examples. The results of the numerical simulations indicate a small difference between the target and synthesized sound fields, and a smaller synthesis error than WFS.

## Acknowledgments

This research was supported by the National Key R&D Program of China under Grant No. 2018YFB1403800.